# Heat flow and the zero of polynomial-a approach to Riemann Hypesis

this is a note after reading the blog:Heat flow and the zero of polynomial.

1.instead of consider the original version:

$\partial_{zz}f(z,t)=\partial_tf(z,t)$.

consider the corresponding “equidistribution version” is also interesting:

$\partial_{zz}f(z,t)=\theta(z,t)\partial_tf(z,t)$,especially $\theta(z,t)=e^{2\pi i\alpha t},\alpha\in R-Q$.

2.

where $f(z)=z^n+a_{n-1}z^{n-1}+...+a_1z+a_0$.

$f(z,t)=\sum_{k=1}^n\sum_{0\leq m\leq k-2,2|k-m}\frac{k!}{m!(k-m)!}z^mt^{k-m}.$

$=\sum_{k=1}^m\sum_{0\leq m\leq k-2,2|k-m}C_k^mt^{k-m})z^mt^{k-m}$

$\sum_{m=0}^{n-2}(\sum_{k=m,2|k-m}^nC_k^mt^{k-m})z^m$.

rescaling:

$F_t:(z_1(t),...,z_n(t))\longrightarrow (\frac{z_1(t)}{t},...,\frac{z_n(t)}{t})$.

$F_t\cdot f(z,t)=\sum_{m=0}^{n-2}(\sum_{k=m,2|k-m}^nC_{k}^mt^{k-n})z^m$.

$\lim_{t\to \infty}F_t\cdot f(z,t)=\sum_{m=0,2|n-m}^{n-2}C_n^mz^m$.(*)

even term $\longrightarrow$ constant.(after renormelization)

odd term $\longrightarrow$ 0(invariant).so at least the sum zeros of is invarient.

by the algebraic fundamental theorem,we have n zero $\{z_1,...,z_n\}$of (*).

until now,we already now if the n zeros is distinct,then because the energy is the energy is the same and the entropy is increase so $\exists T>>0,\forall t_i,t_j>T$,$\{t>T|z_i(t)\} \cap \{t>T|z_j(t)\}=\emptyset$.$\lim_{t\to \infty}|z_i(t)|=\infty$ and $\lim_{t\to \infty}arg(z_i(t))=z_i$.

but how to know the information of the change of direction at “blow up” time?

1.change direction only at $blow up$.

2.energy invariant $\sum_{1\leq i\neq j\leq n}\frac{1}{|x_i-x_j|^2}$.

3.general philosophy

deformation some function under some evolution equation, such like heat equation,wave equation,shrodinger equation.and there is some conversion thing under the equation,and some quantity that could calculate directly such like the trace of spectral.

4.difficultis

this philosophy could generate to the analytic function case,but to make the limit case(I only know how ti deal with this now)coverage.we need very good control on the coefficient.

and to investigate the change of direction at blow up point maybe we need some knowledge about the burid group.

# Fractional uncertain principle

semyon dyatlov的一篇文章

semyon dyatlov的文章https://arxiv.org/pdf/1710.05430.pdf，用fractional uncertainly priciple导出了hyperbolic surface上测地线诱导的zeta函数在$Re(s)>1-\epsilon$只有有限个零点。

2.billiard的传播子，但是这里不一样，文章中的 Schottky groups本质上是对于算子的逆写成一种级数形式其中级数由Schottky group生成，但是对于billiard传播子的情况所有的涉及的热核或者波核的paramatrix不仅仅具备markov性质，起主导作用的却是某种需要X-ray估计的性质，级数和并不是对全空间求而是某种截断了的子空间里面，所以比这个证明要难。建立起billiard的传播子估计是证明inverse spectral problem的重要一步。

3.interval exchange map，但是interval exchange map的结构就好只有这里的traslation，这里有一个像的大小的指数衰减，这是interval exchange map所没有的。interval exchage map可能还需要涉及到一个拆分估计，会更难，这可能可以在interval exchange map上的sarnak猜想有进展。

1.研究极限集$\Lambda_{\Gamma}$的结构，本质上具备某种组合上的树结构，在分式线性变换下树的上方和下方交换，而且对于象有指数级别的衰减，这很像连分数展开中的otrowoski表示。对于分式线性变换和Schottky group作用的体积形变估计是容易得到的。

2.Patterson–Sullivan测度$\mu$是在$\Gamma$作用下的遍历测度，特别的，和 $\Gamma$ 是compatible的，所以变量代换公式成立：

$\int_{\Lambda_{\Gamma}} f(x)d\mu(x) =\int_{\Lambda_{\Gamma}}f(\gamma(x))|\gamma'(x)|_{\delta}^B d\mu(x) \forall \gamma \in \Gamma$

$L_Zf(x) = \sum_{a\in Z,a\to b} f(\gamma_{a′}(x))w_{a′}(x), x ∈ I_b.$

（这一点很重要而且在很多问题中都有用，至少有几个例子：1.有的时候一个椭圆方程的特征值很难做，转而去考察他的发展方程。2.很多数论问题，特别是质数在某些partition集合里面的分布，对于对应的L函数的动力系统的刻画就需要这个方程）

3. 文章中3.1.是bourgain的主要贡献，是所谓的sum-product现象在这里的一个引用，为了得到foriour衰减性估计，我们需要不断拆散区间，实际上在树的每一层上面我们都很清楚怎么把这一层的积分拆散到上一层和下一层，这实际上可以看成一个renormelization方程：

$\int_{\Lambda_{\Gamma}}f d\mu =\int_{\Lambda_{\Gamma}}L_{Z(\tau)}^{2k+1} f d\mu = \sum_{A,B,A\leftrightarrow B}f(\gamma_{A∗B}(x))w_{A∗B}(x)d\mu(x)$.

1中的形变估计(只需要估计一下交叉项带来的误差)告诉我们：$| \int_{}fdμ|^2 ≤C\tau^{(2k−1)\delta}\sum_{A,B,A\leftrightarrow B} |\int_{I_b(A)} e^{iξ\phi(\gamma_{A∗B}(x))}w_{a′_k} (x)d\mu(x)| ^2 +C\tau^2$.

$|\int_{\Lambda_{\Gamma}} f d\mu|^2 ≤ C\tau^{(2k+1)\delta}\sum_A sup_{\eta\in J_{\tau}}| e^{2πi\eta\xi_{1,A} (b_1)···\xi_{k,A} (b_k )}| + C\tau^{\delta/4}$. （*）

$\int_{\Lambda_{\Gamma}}exp ( i\xi\phi(x)) g(x) d\mu(x) ≤ C|\xi|^{−\epsilon_1} \forall \xi, |\xi| > 1$.